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1: 24.20 Tables
§24.20 Tables
2: 24.4 Basic Properties
§24.4(iii) Sums of Powers
3: 27.2 Functions
27.2.6 ϕ k ( n ) = ( m , n ) = 1 m k ,
the sum of the k th powers of the positive integers m n that are relatively prime to n .
27.2.7 ϕ ( n ) = ϕ 0 ( n ) .
27.2.10 σ α ( n ) = d | n d α ,
is the sum of the α th powers of the divisors of n , where the exponent α can be real or complex. …
4: 27.3 Multiplicative Properties
27.3.6 σ α ( n ) = r = 1 ν ( n ) p r α ( 1 + a r ) - 1 p r α - 1 , α 0 .
27.3.7 σ α ( m ) σ α ( n ) = d | ( m , n ) d α σ α ( m n d 2 ) ,
5: 27.7 Lambert Series as Generating Functions
If | x | < 1 , then the quotient x n / ( 1 - x n ) is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: …
27.7.5 n = 1 n α x n 1 - x n = n = 1 σ α ( n ) x n ,
6: 27.6 Divisor Sums
27.6.6 d | n ϕ k ( d ) ( n d ) k = 1 k + 2 k + + n k ,
7: 24.19 Methods of Computation
  • Tanner and Wagstaff (1987) derives a congruence ( mod p ) for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

  • 8: Bibliography
  • T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.
  • 9: 27.4 Euler Products and Dirichlet Series
    27.4.11 n = 1 σ α ( n ) n - s = ζ ( s ) ζ ( s - α ) , s > max ( 1 , 1 + α ) ,
    10: 27.11 Asymptotic Formulas: Partial Sums
    27.11.4 n x σ 1 ( n ) = π 2 12 x 2 + O ( x ln x ) .
    27.11.5 n x σ α ( n ) = ζ ( α + 1 ) α + 1 x α + 1 + O ( x β ) , α > 0 , α 1 , β = max ( 1 , α ) .